LMFDB - Embedded newform 1200.3.bg.b.193.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1200,3,Mod(193,1200)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1200, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 0, 3])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1200.193"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$N$	$=$	$1200 = 2^{4} \cdot 3 \cdot 5^{2}$
Weight:	$k$	$=$	$3$
Character orbit:	$[\chi]$	$=$	1200.bg (of order $4$ , degree $2$ , not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,-16,0,0,0,8,0,0,0,0,0,16,0,0,0,-24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(21)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$32.6976317232$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(i, \sqrt{6})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} + 9$ x^4 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 600)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			193.2
Root			$1.22474 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	1200.193
Dual form			1200.3.bg.b.1057.2

$q$ -expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.22474 + 1.22474i) q^{3} +(-6.44949 + 6.44949i) q^{7} +3.00000i q^{9} +11.7980 q^{11} +(-2.44949 - 2.44949i) q^{13} +(-0.898979 + 0.898979i) q^{17} +33.5959i q^{19} -15.7980 q^{21} +(-21.7980 - 21.7980i) q^{23} +(-3.67423 + 3.67423i) q^{27} -3.59592i q^{29} +25.1918 q^{31} +(14.4495 + 14.4495i) q^{33} +(-9.14643 + 9.14643i) q^{37} -6.00000i q^{39} -60.7878 q^{41} +(22.2020 + 22.2020i) q^{43} +(-4.49490 + 4.49490i) q^{47} -34.1918i q^{49} -2.20204 q^{51} +(-52.0908 - 52.0908i) q^{53} +(-41.1464 + 41.1464i) q^{57} +86.9898i q^{59} -108.384 q^{61} +(-19.3485 - 19.3485i) q^{63} +(-77.8888 + 77.8888i) q^{67} -53.3939i q^{69} +66.7878 q^{71} +(-78.6969 - 78.6969i) q^{73} +(-76.0908 + 76.0908i) q^{77} -66.0000i q^{79} -9.00000 q^{81} +(-84.0908 - 84.0908i) q^{83} +(4.40408 - 4.40408i) q^{87} -9.59592i q^{89} +31.5959 q^{91} +(30.8536 + 30.8536i) q^{93} +(-53.0806 + 53.0806i) q^{97} +35.3939i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q - 16 q^{7} + 8 q^{11} + 16 q^{17} - 24 q^{21} - 48 q^{23} - 56 q^{31} + 48 q^{33} + 32 q^{37} - 8 q^{41} + 128 q^{43} + 80 q^{47} - 48 q^{51} - 32 q^{53} - 96 q^{57} - 120 q^{61} - 48 q^{63} - 96 q^{67}+ \cdots + 160 q^{97}+O(q^{100})$ 4 * q - 16 * q^7 + 8 * q^11 + 16 * q^17 - 24 * q^21 - 48 * q^23 - 56 * q^31 + 48 * q^33 + 32 * q^37 - 8 * q^41 + 128 * q^43 + 80 * q^47 - 48 * q^51 - 32 * q^53 - 96 * q^57 - 120 * q^61 - 48 * q^63 - 96 * q^67 + 32 * q^71 - 256 * q^73 - 128 * q^77 - 36 * q^81 - 160 * q^83 + 96 * q^87 + 48 * q^91 + 192 * q^93 + 160 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times$ .

$n$	$401$	$577$	$751$	$901$
$\chi(n)$	$1$	$e\left(\frac{3}{4}\right)$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$ -expansion $a_n$ , the Satake parameters $\alpha_p$ , and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$ .

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$\alpha_n$			$\theta_n$
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$\alpha_p$			$\theta_p$

$2$		0			0
$2$		0			0
$3$	1.22474	+	1.22474i	0.408248	+	0.408248i
$3$	1.22474	+	1.22474i	0.408248	+	0.408248i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−6.44949	+	6.44949i	−0.921356	+	0.921356i	−0.997125	−	0.0757697i	$-0.975859\pi$
$7$	−6.44949	+	6.44949i	−0.921356	+	0.921356i	0.0757697	+	0.997125i	$0.475859\pi$
$8$		0			0
$9$			3.00000i			0.333333i
$10$		0			0
$11$	11.7980			1.07254			0.536271	−	0.844046i	$-0.319833\pi$
$11$	11.7980			1.07254			0.536271	+	0.844046i	$0.319833\pi$
$12$		0			0
$13$	−2.44949	−	2.44949i	−0.188422	−	0.188422i	0.606591	−	0.795014i	$-0.292536\pi$
$13$	−2.44949	−	2.44949i	−0.188422	−	0.188422i	−0.795014	+	0.606591i	$0.792536\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−0.898979	+	0.898979i	−0.0528811	+	0.0528811i	−0.733053	−	0.680172i	$-0.761905\pi$
$17$	−0.898979	+	0.898979i	−0.0528811	+	0.0528811i	0.680172	+	0.733053i	$0.261905\pi$
$18$		0			0
$19$			33.5959i			1.76821i	0.467292	+	0.884103i	$0.345230\pi$
$19$			33.5959i			1.76821i	−0.467292	+	0.884103i	$0.654770\pi$
$20$		0			0
$21$	−15.7980			−0.752284
$22$		0			0
$23$	−21.7980	−	21.7980i	−0.947737	−	0.947737i	0.0509632	−	0.998701i	$-0.483771\pi$
$23$	−21.7980	−	21.7980i	−0.947737	−	0.947737i	−0.998701	+	0.0509632i	$0.983771\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−3.67423	+	3.67423i	−0.136083	+	0.136083i
$28$		0			0
$29$		−	3.59592i		−	0.123997i	−0.998076	−	0.0619986i	$-0.980253\pi$
$29$		−	3.59592i		−	0.123997i	0.998076	−	0.0619986i	$-0.0197474\pi$
$30$		0			0
$31$	25.1918			0.812640			0.406320	−	0.913731i	$-0.366812\pi$
$31$	25.1918			0.812640			0.406320	+	0.913731i	$0.366812\pi$
$32$		0			0
$33$	14.4495	+	14.4495i	0.437863	+	0.437863i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−9.14643	+	9.14643i	−0.247201	+	0.247201i	−0.819821	−	0.572620i	$-0.805927\pi$
$37$	−9.14643	+	9.14643i	−0.247201	+	0.247201i	0.572620	+	0.819821i	$0.305927\pi$
$38$		0			0
$39$		−	6.00000i		−	0.153846i
$40$		0			0
$41$	−60.7878			−1.48263			−0.741314	−	0.671158i	$-0.765797\pi$
$41$	−60.7878			−1.48263			−0.741314	+	0.671158i	$0.765797\pi$
$42$		0			0
$43$	22.2020	+	22.2020i	0.516327	+	0.516327i	0.916458	−	0.400131i	$-0.131035\pi$
$43$	22.2020	+	22.2020i	0.516327	+	0.516327i	−0.400131	+	0.916458i	$0.631035\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−4.49490	+	4.49490i	−0.0956361	+	0.0956361i	−0.753306	−	0.657670i	$-0.771542\pi$
$47$	−4.49490	+	4.49490i	−0.0956361	+	0.0956361i	0.657670	+	0.753306i	$0.271542\pi$
$48$		0			0
$49$		−	34.1918i		−	0.697793i
$50$		0			0
$51$	−2.20204			−0.0431773
$52$		0			0
$53$	−52.0908	−	52.0908i	−0.982846	−	0.982846i	0.0170098	−	0.999855i	$-0.494585\pi$
$53$	−52.0908	−	52.0908i	−0.982846	−	0.982846i	−0.999855	+	0.0170098i	$0.994585\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−41.1464	+	41.1464i	−0.721867	+	0.721867i
$58$		0			0
$59$			86.9898i			1.47440i	0.675673	+	0.737202i	$0.263853\pi$
$59$			86.9898i			1.47440i	−0.675673	+	0.737202i	$0.736147\pi$
$60$		0			0
$61$	−108.384			−1.77678			−0.888391	−	0.459088i	$-0.848176\pi$
$61$	−108.384			−1.77678			−0.888391	+	0.459088i	$0.848176\pi$
$62$		0			0
$63$	−19.3485	−	19.3485i	−0.307119	−	0.307119i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−77.8888	+	77.8888i	−1.16252	+	1.16252i	−0.178597	+	0.983922i	$0.557156\pi$
$67$	−77.8888	+	77.8888i	−1.16252	+	1.16252i	−0.983922	+	0.178597i	$0.942844\pi$
$68$		0			0
$69$		−	53.3939i		−	0.773824i
$70$		0			0
$71$	66.7878			0.940673			0.470336	−	0.882487i	$-0.344133\pi$
$71$	66.7878			0.940673			0.470336	+	0.882487i	$0.344133\pi$
$72$		0			0
$73$	−78.6969	−	78.6969i	−1.07804	−	1.07804i	−0.996685	−	0.0813551i	$-0.974075\pi$
$73$	−78.6969	−	78.6969i	−1.07804	−	1.07804i	−0.0813551	−	0.996685i	$-0.525925\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−76.0908	+	76.0908i	−0.988192	+	0.988192i
$78$		0			0
$79$		−	66.0000i		−	0.835443i	−0.908575	−	0.417722i	$-0.862829\pi$
$79$		−	66.0000i		−	0.835443i	0.908575	−	0.417722i	$-0.137171\pi$
$80$		0			0
$81$	−9.00000			−0.111111
$82$		0			0
$83$	−84.0908	−	84.0908i	−1.01314	−	1.01314i	−0.999912	−	0.0132299i	$-0.995789\pi$
$83$	−84.0908	−	84.0908i	−1.01314	−	1.01314i	−0.0132299	−	0.999912i	$-0.504211\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	4.40408	−	4.40408i	0.0506216	−	0.0506216i
$88$		0			0
$89$		−	9.59592i		−	0.107819i	−0.998546	−	0.0539097i	$-0.982832\pi$
$89$		−	9.59592i		−	0.107819i	0.998546	−	0.0539097i	$-0.0171683\pi$
$90$		0			0
$91$	31.5959			0.347208
$92$		0			0
$93$	30.8536	+	30.8536i	0.331759	+	0.331759i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−53.0806	+	53.0806i	−0.547223	+	0.547223i	−0.925637	−	0.378414i	$-0.876470\pi$
$97$	−53.0806	+	53.0806i	−0.547223	+	0.547223i	0.378414	+	0.925637i	$0.376470\pi$
$98$		0			0
$99$			35.3939i			0.357514i
$100$		0			0
$101$	155.192			1.53655			0.768276	−	0.640118i	$-0.221115\pi$
$101$	155.192			1.53655			0.768276	+	0.640118i	$0.221115\pi$
$102$		0			0
$103$	42.3587	+	42.3587i	0.411249	+	0.411249i	0.882174	−	0.470924i	$-0.156080\pi$
$103$	42.3587	+	42.3587i	0.411249	+	0.411249i	−0.470924	+	0.882174i	$0.656080\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−29.8888	+	29.8888i	−0.279334	+	0.279334i	−0.832843	−	0.553509i	$-0.813288\pi$
$107$	−29.8888	+	29.8888i	−0.279334	+	0.279334i	0.553509	+	0.832843i	$0.313288\pi$
$108$		0			0
$109$			17.1918i			0.157723i	0.996886	+	0.0788616i	$0.0251285\pi$
$109$			17.1918i			0.157723i	−0.996886	+	0.0788616i	$0.974871\pi$
$110$		0			0
$111$	−22.4041			−0.201839
$112$		0			0
$113$	134.293	+	134.293i	1.18843	+	1.18843i	0.977501	+	0.210932i	$0.0676497\pi$
$113$	134.293	+	134.293i	1.18843	+	1.18843i	0.210932	+	0.977501i	$0.432350\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	7.34847	−	7.34847i	0.0628074	−	0.0628074i
$118$		0			0
$119$		−	11.5959i		−	0.0974447i
$120$		0			0
$121$	18.1918			0.150346
$122$		0			0
$123$	−74.4495	−	74.4495i	−0.605280	−	0.605280i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−153.237	+	153.237i	−1.20659	+	1.20659i	−0.234469	+	0.972124i	$0.575335\pi$
$127$	−153.237	+	153.237i	−1.20659	+	1.20659i	−0.972124	+	0.234469i	$0.924665\pi$
$128$		0			0
$129$			54.3837i			0.421579i
$130$		0			0
$131$	196.606			1.50081			0.750405	−	0.660978i	$-0.229858\pi$
$131$	196.606			1.50081			0.750405	+	0.660978i	$0.229858\pi$
$132$		0			0
$133$	−216.677	−	216.677i	−1.62915	−	1.62915i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−81.0806	+	81.0806i	−0.591829	+	0.591829i	−0.938125	−	0.346296i	$-0.887439\pi$
$137$	−81.0806	+	81.0806i	−0.591829	+	0.591829i	0.346296	+	0.938125i	$0.387439\pi$
$138$		0			0
$139$		−	37.1918i		−	0.267567i	−0.991011	−	0.133784i	$-0.957287\pi$
$139$		−	37.1918i		−	0.267567i	0.991011	−	0.133784i	$-0.0427127\pi$
$140$		0			0
$141$	−11.0102			−0.0780866
$142$		0			0
$143$	−28.8990	−	28.8990i	−0.202091	−	0.202091i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	41.8763	−	41.8763i	0.284873	−	0.284873i
$148$		0			0
$149$			86.7878i			0.582468i	0.956652	+	0.291234i	$0.0940658\pi$
$149$			86.7878i			0.582468i	−0.956652	+	0.291234i	$0.905934\pi$
$150$		0			0
$151$	−266.767			−1.76667			−0.883336	−	0.468741i	$-0.844708\pi$
$151$	−266.767			−1.76667			−0.883336	+	0.468741i	$0.844708\pi$
$152$		0			0
$153$	−2.69694	−	2.69694i	−0.0176270	−	0.0176270i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	203.439	−	203.439i	1.29579	−	1.29579i	0.364645	−	0.931147i	$-0.381190\pi$
$157$	203.439	−	203.439i	1.29579	−	1.29579i	0.931147	−	0.364645i	$-0.118810\pi$
$158$		0			0
$159$		−	127.596i		−	0.802490i
$160$		0			0
$161$	281.171			1.74641
$162$		0			0
$163$	29.2122	+	29.2122i	0.179216	+	0.179216i	0.791014	−	0.611798i	$-0.209553\pi$
$163$	29.2122	+	29.2122i	0.179216	+	0.179216i	−0.611798	+	0.791014i	$0.709553\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−116.182	+	116.182i	−0.695698	+	0.695698i	−0.963480	−	0.267781i	$-0.913710\pi$
$167$	−116.182	+	116.182i	−0.695698	+	0.695698i	0.267781	+	0.963480i	$0.413710\pi$
$168$		0			0
$169$		−	157.000i		−	0.928994i
$170$		0			0
$171$	−100.788			−0.589402
$172$		0			0
$173$	−60.2724	−	60.2724i	−0.348396	−	0.348396i	0.511116	−	0.859512i	$-0.329232\pi$
$173$	−60.2724	−	60.2724i	−0.348396	−	0.348396i	−0.859512	+	0.511116i	$0.829232\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−106.540	+	106.540i	−0.601923	+	0.601923i
$178$		0			0
$179$		−	176.161i		−	0.984141i	−0.870555	−	0.492070i	$-0.836240\pi$
$179$		−	176.161i		−	0.984141i	0.870555	−	0.492070i	$-0.163760\pi$
$180$		0			0
$181$	161.959			0.894802			0.447401	−	0.894333i	$-0.352350\pi$
$181$	161.959			0.894802			0.447401	+	0.894333i	$0.352350\pi$
$182$		0			0
$183$	−132.742	−	132.742i	−0.725368	−	0.725368i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−10.6061	+	10.6061i	−0.0567172	+	0.0567172i
$188$		0			0
$189$		−	47.3939i		−	0.250761i
$190$		0			0
$191$	−250.747			−1.31281			−0.656406	−	0.754408i	$-0.727924\pi$
$191$	−250.747			−1.31281			−0.656406	+	0.754408i	$0.727924\pi$
$192$		0			0
$193$	−8.62653	−	8.62653i	−0.0446971	−	0.0446971i	0.684405	−	0.729102i	$-0.260062\pi$
$193$	−8.62653	−	8.62653i	−0.0446971	−	0.0446971i	−0.729102	+	0.684405i	$0.760062\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−182.202	+	182.202i	−0.924883	+	0.924883i	−0.997369	−	0.0724860i	$-0.976907\pi$
$197$	−182.202	+	182.202i	−0.924883	+	0.924883i	0.0724860	+	0.997369i	$0.476907\pi$
$198$		0			0
$199$		−	81.1918i		−	0.407999i	−0.978971	−	0.204000i	$-0.934606\pi$
$199$		−	81.1918i		−	0.407999i	0.978971	−	0.204000i	$-0.0653941\pi$
$200$		0			0
$201$	−190.788			−0.949193
$202$		0			0
$203$	23.1918	+	23.1918i	0.114245	+	0.114245i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	65.3939	−	65.3939i	0.315912	−	0.315912i
$208$		0			0
$209$			396.363i			1.89647i
$210$		0			0
$211$	−199.576			−0.945855			−0.472928	−	0.881101i	$-0.656803\pi$
$211$	−199.576			−0.945855			−0.472928	+	0.881101i	$0.656803\pi$
$212$		0			0
$213$	81.7980	+	81.7980i	0.384028	+	0.384028i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−162.474	+	162.474i	−0.748730	+	0.748730i
$218$		0			0
$219$		−	192.767i		−	0.880216i
$220$		0			0
$221$	4.40408			0.0199280
$222$		0			0
$223$	134.944	+	134.944i	0.605132	+	0.605132i	0.941670	−	0.336538i	$-0.109256\pi$
$223$	134.944	+	134.944i	0.605132	+	0.605132i	−0.336538	+	0.941670i	$0.609256\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	60.8990	−	60.8990i	0.268277	−	0.268277i	−0.560128	−	0.828406i	$-0.689248\pi$
$227$	60.8990	−	60.8990i	0.268277	−	0.268277i	0.828406	+	0.560128i	$0.189248\pi$
$228$		0			0
$229$			138.767i			0.605971i	0.952995	+	0.302985i	$0.0979834\pi$
$229$			138.767i			0.605971i	−0.952995	+	0.302985i	$0.902017\pi$
$230$		0			0
$231$	−186.384			−0.806856
$232$		0			0
$233$	254.656	+	254.656i	1.09294	+	1.09294i	0.995213	+	0.0977319i	$0.0311588\pi$
$233$	254.656	+	254.656i	1.09294	+	1.09294i	0.0977319	+	0.995213i	$0.468841\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	80.8332	−	80.8332i	0.341068	−	0.341068i
$238$		0			0
$239$			422.788i			1.76899i	0.466553	+	0.884493i	$0.345496\pi$
$239$			422.788i			1.76899i	−0.466553	+	0.884493i	$0.654504\pi$
$240$		0			0
$241$	10.7673			0.0446778			0.0223389	−	0.999750i	$-0.492889\pi$
$241$	10.7673			0.0446778			0.0223389	+	0.999750i	$0.492889\pi$
$242$		0			0
$243$	−11.0227	−	11.0227i	−0.0453609	−	0.0453609i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	82.2929	−	82.2929i	0.333169	−	0.333169i
$248$		0			0
$249$		−	205.980i		−	0.827227i
$250$		0			0
$251$	177.818			0.708440			0.354220	−	0.935162i	$-0.384746\pi$
$251$	177.818			0.708440			0.354220	+	0.935162i	$0.384746\pi$
$252$		0			0
$253$	−257.171	−	257.171i	−1.01649	−	1.01649i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	142.293	−	142.293i	0.553669	−	0.553669i	−0.373829	−	0.927498i	$-0.621955\pi$
$257$	142.293	−	142.293i	0.553669	−	0.553669i	0.927498	+	0.373829i	$0.121955\pi$
$258$		0			0
$259$		−	117.980i		−	0.455520i
$260$		0			0
$261$	10.7878			0.0413324
$262$		0			0
$263$	−36.0000	−	36.0000i	−0.136882	−	0.136882i	0.635346	−	0.772228i	$-0.280858\pi$
$263$	−36.0000	−	36.0000i	−0.136882	−	0.136882i	−0.772228	+	0.635346i	$0.780858\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	11.7526	−	11.7526i	0.0440170	−	0.0440170i
$268$		0			0
$269$			38.8286i			0.144344i	0.997392	+	0.0721721i	$0.0229931\pi$
$269$			38.8286i			0.144344i	−0.997392	+	0.0721721i	$0.977007\pi$
$270$		0			0
$271$	11.5755			0.0427141			0.0213570	−	0.999772i	$-0.493201\pi$
$271$	11.5755			0.0427141			0.0213570	+	0.999772i	$0.493201\pi$
$272$		0			0
$273$	38.6969	+	38.6969i	0.141747	+	0.141747i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	282.499	−	282.499i	1.01985	−	1.01985i	0.0200549	−	0.999799i	$-0.493616\pi$
$277$	282.499	−	282.499i	1.01985	−	1.01985i	0.999799	−	0.0200549i	$-0.00638409\pi$
$278$		0			0
$279$			75.5755i			0.270880i
$280$		0			0
$281$	432.384			1.53873			0.769366	−	0.638808i	$-0.220572\pi$
$281$	432.384			1.53873			0.769366	+	0.638808i	$0.220572\pi$
$282$		0			0
$283$	223.868	+	223.868i	0.791054	+	0.791054i	0.981666	−	0.190611i	$-0.0610470\pi$
$283$	223.868	+	223.868i	0.791054	+	0.791054i	−0.190611	+	0.981666i	$0.561047\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	392.050	−	392.050i	1.36603	−	1.36603i
$288$		0			0
$289$			287.384i			0.994407i
$290$		0			0
$291$	−130.020			−0.446806
$292$		0			0
$293$	−196.767	−	196.767i	−0.671561	−	0.671561i	0.286515	−	0.958076i	$-0.407503\pi$
$293$	−196.767	−	196.767i	−0.671561	−	0.671561i	−0.958076	+	0.286515i	$0.907503\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−43.3485	+	43.3485i	−0.145954	+	0.145954i
$298$		0			0
$299$			106.788i			0.357150i
$300$		0			0
$301$	−286.384			−0.951441
$302$		0			0
$303$	190.070	+	190.070i	0.627295	+	0.627295i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−141.707	+	141.707i	−0.461587	+	0.461587i	−0.899175	−	0.437589i	$-0.855833\pi$
$307$	−141.707	+	141.707i	−0.461587	+	0.461587i	0.437589	+	0.899175i	$0.355833\pi$
$308$		0			0
$309$			103.757i			0.335784i
$310$		0			0
$311$	411.555			1.32333			0.661664	−	0.749800i	$-0.269850\pi$
$311$	411.555			1.32333			0.661664	+	0.749800i	$0.269850\pi$
$312$		0			0
$313$	365.939	+	365.939i	1.16913	+	1.16913i	0.982413	+	0.186720i	$0.0597858\pi$
$313$	365.939	+	365.939i	1.16913	+	1.16913i	0.186720	+	0.982413i	$0.440214\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	338.343	−	338.343i	1.06733	−	1.06733i	0.0697641	−	0.997564i	$-0.477775\pi$
$317$	338.343	−	338.343i	1.06733	−	1.06733i	0.997564	−	0.0697641i	$-0.0222247\pi$
$318$		0			0
$319$		−	42.4245i		−	0.132992i
$320$		0			0
$321$	−73.2122			−0.228076
$322$		0			0
$323$	−30.2020	−	30.2020i	−0.0935048	−	0.0935048i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−21.0556	+	21.0556i	−0.0643903	+	0.0643903i
$328$		0			0
$329$		−	57.9796i		−	0.176230i
$330$		0			0
$331$	97.5551			0.294728			0.147364	−	0.989082i	$-0.452921\pi$
$331$	97.5551			0.294728			0.147364	+	0.989082i	$0.452921\pi$
$332$		0			0
$333$	−27.4393	−	27.4393i	−0.0824003	−	0.0824003i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	72.4949	−	72.4949i	0.215118	−	0.215118i	−0.591319	−	0.806438i	$-0.701393\pi$
$337$	72.4949	−	72.4949i	0.215118	−	0.215118i	0.806438	+	0.591319i	$0.201393\pi$
$338$		0			0
$339$			328.949i			0.970351i
$340$		0			0
$341$	297.212			0.871590
$342$		0			0
$343$	−95.5051	−	95.5051i	−0.278441	−	0.278441i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	40.8582	−	40.8582i	0.117747	−	0.117747i	−0.645778	−	0.763525i	$-0.723467\pi$
$347$	40.8582	−	40.8582i	0.117747	−	0.117747i	0.763525	+	0.645778i	$0.223467\pi$
$348$		0			0
$349$		−	332.343i		−	0.952272i	−0.879372	−	0.476136i	$-0.842037\pi$
$349$		−	332.343i		−	0.952272i	0.879372	−	0.476136i	$-0.157963\pi$
$350$		0			0
$351$	18.0000			0.0512821
$352$		0			0
$353$	260.232	+	260.232i	0.737200	+	0.737200i	0.972035	−	0.234835i	$-0.0754550\pi$
$353$	260.232	+	260.232i	0.737200	+	0.737200i	−0.234835	+	0.972035i	$0.575455\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	14.2020	−	14.2020i	0.0397816	−	0.0397816i
$358$		0			0
$359$			332.000i			0.924791i	0.886674	+	0.462396i	$0.153010\pi$
$359$			332.000i			0.924791i	−0.886674	+	0.462396i	$0.846990\pi$
$360$		0			0
$361$	−767.686			−2.12655
$362$		0			0
$363$	22.2804	+	22.2804i	0.0613784	+	0.0613784i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	210.409	−	210.409i	0.573321	−	0.573321i	−0.359734	−	0.933055i	$-0.617133\pi$
$367$	210.409	−	210.409i	0.573321	−	0.573321i	0.933055	+	0.359734i	$0.117133\pi$
$368$		0			0
$369$		−	182.363i		−	0.494209i
$370$		0			0
$371$	671.918			1.81110
$372$		0			0
$373$	222.672	+	222.672i	0.596976	+	0.596976i	0.939507	−	0.342531i	$-0.111284\pi$
$373$	222.672	+	222.672i	0.596976	+	0.596976i	−0.342531	+	0.939507i	$0.611284\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−8.80816	+	8.80816i	−0.0233638	+	0.0233638i
$378$		0			0
$379$			222.767i			0.587777i	0.955840	+	0.293888i	$0.0949494\pi$
$379$			222.767i			0.587777i	−0.955840	+	0.293888i	$0.905051\pi$
$380$		0			0
$381$	−375.353			−0.985179
$382$		0			0
$383$	−120.717	−	120.717i	−0.315189	−	0.315189i	0.531727	−	0.846916i	$-0.321543\pi$
$383$	−120.717	−	120.717i	−0.315189	−	0.315189i	−0.846916	+	0.531727i	$0.821543\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−66.6061	+	66.6061i	−0.172109	+	0.172109i
$388$		0			0
$389$			80.8082i			0.207733i	0.994591	+	0.103867i	$0.0331215\pi$
$389$			80.8082i			0.207733i	−0.994591	+	0.103867i	$0.966879\pi$
$390$		0			0
$391$	39.1918			0.100235
$392$		0			0
$393$	240.792	+	240.792i	0.612703	+	0.612703i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−21.9954	+	21.9954i	−0.0554041	+	0.0554041i	−0.734266	−	0.678862i	$-0.762473\pi$
$397$	−21.9954	+	21.9954i	−0.0554041	+	0.0554041i	0.678862	+	0.734266i	$0.262473\pi$
$398$		0			0
$399$		−	530.747i		−	1.33019i
$400$		0			0
$401$	−267.131			−0.666161			−0.333081	−	0.942898i	$-0.608088\pi$
$401$	−267.131			−0.666161			−0.333081	+	0.942898i	$0.608088\pi$
$402$		0			0
$403$	−61.7071	−	61.7071i	−0.153119	−	0.153119i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−107.909	+	107.909i	−0.265133	+	0.265133i
$408$		0			0
$409$		−	510.767i		−	1.24882i	−0.781097	−	0.624410i	$-0.785339\pi$
$409$		−	510.767i		−	1.24882i	0.781097	−	0.624410i	$-0.214661\pi$
$410$		0			0
$411$	−198.606			−0.483227
$412$		0			0
$413$	−561.040	−	561.040i	−1.35845	−	1.35845i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	45.5505	−	45.5505i	0.109234	−	0.109234i
$418$		0			0
$419$			40.1204i			0.0957527i	0.998853	+	0.0478764i	$0.0152454\pi$
$419$			40.1204i			0.0957527i	−0.998853	+	0.0478764i	$0.984755\pi$
$420$		0			0
$421$	−132.343			−0.314354			−0.157177	−	0.987570i	$-0.550239\pi$
$421$	−132.343			−0.314354			−0.157177	+	0.987570i	$0.550239\pi$
$422$		0			0
$423$	−13.4847	−	13.4847i	−0.0318787	−	0.0318787i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	699.019	−	699.019i	1.63705	−	1.63705i
$428$		0			0
$429$		−	70.7878i		−	0.165006i
$430$		0			0
$431$	510.343			1.18409			0.592045	−	0.805905i	$-0.298321\pi$
$431$	510.343			1.18409			0.592045	+	0.805905i	$0.298321\pi$
$432$		0			0
$433$	443.828	+	443.828i	1.02501	+	1.02501i	0.999679	+	0.0253266i	$0.00806258\pi$
$433$	443.828	+	443.828i	1.02501	+	1.02501i	0.0253266	+	0.999679i	$0.491937\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	732.322	−	732.322i	1.67580	−	1.67580i
$438$		0			0
$439$		−	306.808i		−	0.698880i	−0.936959	−	0.349440i	$-0.886372\pi$
$439$		−	306.808i		−	0.698880i	0.936959	−	0.349440i	$-0.113628\pi$
$440$		0			0
$441$	102.576			0.232598
$442$		0			0
$443$	443.646	+	443.646i	1.00146	+	1.00146i	0.999999	+	0.00145911i	$0.000464448\pi$
$443$	443.646	+	443.646i	1.00146	+	1.00146i	0.00145911	+	0.999999i	$0.499536\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−106.293	+	106.293i	−0.237792	+	0.237792i
$448$		0			0
$449$		−	230.727i		−	0.513868i	−0.966429	−	0.256934i	$-0.917288\pi$
$449$		−	230.727i		−	0.513868i	0.966429	−	0.256934i	$-0.0827122\pi$
$450$		0			0
$451$	−717.171			−1.59018
$452$		0			0
$453$	−326.722	−	326.722i	−0.721240	−	0.721240i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	27.2827	−	27.2827i	0.0596995	−	0.0596995i	−0.676627	−	0.736326i	$-0.736559\pi$
$457$	27.2827	−	27.2827i	0.0596995	−	0.0596995i	0.736326	+	0.676627i	$0.236559\pi$
$458$		0			0
$459$		−	6.60612i		−	0.0143924i
$460$		0			0
$461$	−256.322			−0.556014			−0.278007	−	0.960579i	$-0.589674\pi$
$461$	−256.322			−0.556014			−0.278007	+	0.960579i	$0.589674\pi$
$462$		0			0
$463$	173.196	+	173.196i	0.374074	+	0.374074i	0.868959	−	0.494884i	$-0.164790\pi$
$463$	173.196	+	173.196i	0.374074	+	0.374074i	−0.494884	+	0.868959i	$0.664790\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	285.707	−	285.707i	0.611793	−	0.611793i	−0.331620	−	0.943413i	$-0.607595\pi$
$467$	285.707	−	285.707i	0.611793	−	0.611793i	0.943413	+	0.331620i	$0.107595\pi$
$468$		0			0
$469$		−	1004.69i		−	2.14219i
$470$		0			0
$471$	498.322			1.05801
$472$		0			0
$473$	261.939	+	261.939i	0.553782	+	0.553782i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	156.272	−	156.272i	0.327615	−	0.327615i
$478$		0			0
$479$			356.404i			0.744059i	0.928221	+	0.372029i	$0.121338\pi$
$479$			356.404i			0.744059i	−0.928221	+	0.372029i	$0.878662\pi$
$480$		0			0
$481$	44.8082			0.0931563
$482$		0			0
$483$	344.363	+	344.363i	0.712967	+	0.712967i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−55.6209	+	55.6209i	−0.114211	+	0.114211i	−0.761903	−	0.647691i	$-0.775735\pi$
$487$	−55.6209	+	55.6209i	−0.114211	+	0.114211i	0.647691	+	0.761903i	$0.275735\pi$
$488$		0			0
$489$			71.5551i			0.146329i
$490$		0			0
$491$	−106.141			−0.216173			−0.108086	−	0.994142i	$-0.534472\pi$
$491$	−106.141			−0.216173			−0.108086	+	0.994142i	$0.534472\pi$
$492$		0			0
$493$	3.23266	+	3.23266i	0.00655711	+	0.00655711i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−430.747	+	430.747i	−0.866694	+	0.866694i
$498$		0			0
$499$			298.363i			0.597922i	0.954265	+	0.298961i	$0.0966401\pi$
$499$			298.363i			0.597922i	−0.954265	+	0.298961i	$0.903360\pi$
$500$		0			0
$501$	−284.586			−0.568035
$502$		0			0
$503$	655.101	+	655.101i	1.30239	+	1.30239i	0.926778	+	0.375610i	$0.122567\pi$
$503$	655.101	+	655.101i	1.30239	+	1.30239i	0.375610	+	0.926778i	$0.377433\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	192.285	−	192.285i	0.379260	−	0.379260i
$508$		0			0
$509$			683.110i			1.34206i	0.741429	+	0.671032i	$0.234149\pi$
$509$			683.110i			1.34206i	−0.741429	+	0.671032i	$0.765851\pi$
$510$		0			0
$511$	1015.11			1.98652
$512$		0			0
$513$	−123.439	−	123.439i	−0.240622	−	0.240622i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−53.0306	+	53.0306i	−0.102574	+	0.102574i
$518$		0			0
$519$		−	147.637i		−	0.284464i
$520$		0			0
$521$	−6.76734			−0.0129891			−0.00649457	−	0.999979i	$-0.502067\pi$
$521$	−6.76734			−0.0129891			−0.00649457	+	0.999979i	$0.502067\pi$
$522$		0			0
$523$	−600.990	−	600.990i	−1.14912	−	1.14912i	−0.986726	−	0.162394i	$-0.948078\pi$
$523$	−600.990	−	600.990i	−1.14912	−	1.14912i	−0.162394	−	0.986726i	$-0.551922\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−22.6469	+	22.6469i	−0.0429733	+	0.0429733i
$528$		0			0
$529$			421.302i			0.796412i
$530$		0			0
$531$	−260.969			−0.491468
$532$		0			0
$533$	148.899	+	148.899i	0.279360	+	0.279360i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	215.753	−	215.753i	0.401774	−	0.401774i
$538$		0			0
$539$		−	403.394i		−	0.748412i
$540$		0			0
$541$	177.110			0.327376			0.163688	−	0.986512i	$-0.447661\pi$
$541$	177.110			0.327376			0.163688	+	0.986512i	$0.447661\pi$
$542$		0			0
$543$	198.359	+	198.359i	0.365301	+	0.365301i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−35.3326	+	35.3326i	−0.0645935	+	0.0645935i	−0.738666	−	0.674072i	$-0.764544\pi$
$547$	−35.3326	+	35.3326i	−0.0645935	+	0.0645935i	0.674072	+	0.738666i	$0.264544\pi$
$548$		0			0
$549$		−	325.151i		−	0.592261i
$550$		0			0
$551$	120.808			0.219253
$552$		0			0
$553$	425.666	+	425.666i	0.769740	+	0.769740i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−121.930	+	121.930i	−0.218904	+	0.218904i	−0.808037	−	0.589132i	$-0.799470\pi$
$557$	−121.930	+	121.930i	−0.218904	+	0.218904i	0.589132	+	0.808037i	$0.299470\pi$
$558$		0			0
$559$		−	108.767i		−	0.194575i
$560$		0			0
$561$	−25.9796			−0.0463094
$562$		0			0
$563$	−436.636	−	436.636i	−0.775552	−	0.775552i	0.203519	−	0.979071i	$-0.434762\pi$
$563$	−436.636	−	436.636i	−0.775552	−	0.775552i	−0.979071	+	0.203519i	$0.934762\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	58.0454	−	58.0454i	0.102373	−	0.102373i
$568$		0			0
$569$		−	394.363i		−	0.693081i	−0.938035	−	0.346541i	$-0.887356\pi$
$569$		−	394.363i		−	0.693081i	0.938035	−	0.346541i	$-0.112644\pi$
$570$		0			0
$571$	−32.0612			−0.0561493			−0.0280746	−	0.999606i	$-0.508938\pi$
$571$	−32.0612			−0.0561493			−0.0280746	+	0.999606i	$0.508938\pi$
$572$		0			0
$573$	−307.101	−	307.101i	−0.535953	−	0.535953i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−545.848	+	545.848i	−0.946010	+	0.946010i	−0.998615	−	0.0526051i	$-0.983248\pi$
$577$	−545.848	+	545.848i	−0.946010	+	0.946010i	0.0526051	+	0.998615i	$0.483248\pi$
$578$		0			0
$579$		−	21.1306i		−	0.0364950i
$580$		0			0
$581$	1084.69			1.86693
$582$		0			0
$583$	−614.565	−	614.565i	−1.05414	−	1.05414i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−307.728	+	307.728i	−0.524238	+	0.524238i	−0.918848	−	0.394611i	$-0.870879\pi$
$587$	−307.728	+	307.728i	−0.524238	+	0.524238i	0.394611	+	0.918848i	$0.370879\pi$
$588$		0			0
$589$			846.343i			1.43691i
$590$		0			0
$591$	−446.302			−0.755164
$592$		0			0
$593$	−228.940	−	228.940i	−0.386070	−	0.386070i	0.487213	−	0.873283i	$-0.338014\pi$
$593$	−228.940	−	228.940i	−0.386070	−	0.386070i	−0.873283	+	0.487213i	$0.838014\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	99.4393	−	99.4393i	0.166565	−	0.166565i
$598$		0			0
$599$			345.657i			0.577057i	0.957471	+	0.288529i	$0.0931660\pi$
$599$			345.657i			0.577057i	−0.957471	+	0.288529i	$0.906834\pi$
$600$		0			0
$601$	893.069			1.48597			0.742986	−	0.669307i	$-0.233409\pi$
$601$	893.069			1.48597			0.742986	+	0.669307i	$0.233409\pi$
$602$		0			0
$603$	−233.666	−	233.666i	−0.387506	−	0.387506i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	560.974	−	560.974i	0.924175	−	0.924175i	−0.0731466	−	0.997321i	$-0.523304\pi$
$607$	560.974	−	560.974i	0.924175	−	0.924175i	0.997321	+	0.0731466i	$0.0233041\pi$
$608$		0			0
$609$			56.8082i			0.0932811i
$610$		0			0
$611$	22.0204			0.0360400
$612$		0			0
$613$	−439.448	−	439.448i	−0.716882	−	0.716882i	0.251084	−	0.967965i	$-0.419213\pi$
$613$	−439.448	−	439.448i	−0.716882	−	0.716882i	−0.967965	+	0.251084i	$0.919213\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−199.201	+	199.201i	−0.322854	+	0.322854i	−0.849861	−	0.527007i	$-0.823314\pi$
$617$	−199.201	+	199.201i	−0.322854	+	0.322854i	0.527007	+	0.849861i	$0.323314\pi$
$618$		0			0
$619$			121.151i			0.195721i	0.995200	+	0.0978603i	$0.0311998\pi$
$619$			121.151i			0.195721i	−0.995200	+	0.0978603i	$0.968800\pi$
$620$		0			0
$621$	160.182			0.257941
$622$		0			0
$623$	61.8888	+	61.8888i	0.0993399	+	0.0993399i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−485.444	+	485.444i	−0.774233	+	0.774233i
$628$		0			0
$629$		−	16.4449i		−	0.0261445i
$630$		0			0
$631$	−247.494			−0.392225			−0.196112	−	0.980581i	$-0.562832\pi$
$631$	−247.494			−0.392225			−0.196112	+	0.980581i	$0.562832\pi$
$632$		0			0
$633$	−244.429	−	244.429i	−0.386144	−	0.386144i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−83.7526	+	83.7526i	−0.131480	+	0.131480i
$638$		0			0
$639$			200.363i			0.313558i
$640$		0			0
$641$	70.6857			0.110274			0.0551371	−	0.998479i	$-0.482440\pi$
$641$	70.6857			0.110274			0.0551371	+	0.998479i	$0.482440\pi$
$642$		0			0
$643$	−231.373	−	231.373i	−0.359834	−	0.359834i	0.503917	−	0.863752i	$-0.331892\pi$
$643$	−231.373	−	231.373i	−0.359834	−	0.359834i	−0.863752	+	0.503917i	$0.831892\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−500.958	+	500.958i	−0.774278	+	0.774278i	−0.978851	−	0.204573i	$-0.934419\pi$
$647$	−500.958	+	500.958i	−0.774278	+	0.774278i	0.204573	+	0.978851i	$0.434419\pi$
$648$		0			0
$649$			1026.30i			1.58136i
$650$		0			0
$651$	−397.980			−0.611336
$652$		0			0
$653$	261.031	+	261.031i	0.399741	+	0.399741i	0.878142	−	0.478401i	$-0.158783\pi$
$653$	261.031	+	261.031i	0.399741	+	0.399741i	−0.478401	+	0.878142i	$0.658783\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	236.091	−	236.091i	0.359347	−	0.359347i
$658$		0			0
$659$		−	800.524i		−	1.21476i	−0.794413	−	0.607378i	$-0.792221\pi$
$659$		−	800.524i		−	1.21476i	0.794413	−	0.607378i	$-0.207779\pi$
$660$		0			0
$661$	−313.192			−0.473815			−0.236908	−	0.971532i	$-0.576134\pi$
$661$	−313.192			−0.473815			−0.236908	+	0.971532i	$0.576134\pi$
$662$		0			0
$663$	5.39388	+	5.39388i	0.00813556	+	0.00813556i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−78.3837	+	78.3837i	−0.117517	+	0.117517i
$668$		0			0
$669$			330.545i			0.494088i
$670$		0			0
$671$	−1278.71			−1.90567
$672$		0			0
$673$	688.677	+	688.677i	1.02329	+	1.02329i	0.999722	+	0.0235713i	$0.00750369\pi$
$673$	688.677	+	688.677i	1.02329	+	1.02329i	0.0235713	+	0.999722i	$0.492496\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	660.767	−	660.767i	0.976023	−	0.976023i	−0.0236965	−	0.999719i	$-0.507544\pi$
$677$	660.767	−	660.767i	0.976023	−	0.976023i	0.999719	+	0.0236965i	$0.00754355\pi$
$678$		0			0
$679$		−	684.686i		−	1.00837i
$680$		0			0
$681$	149.171			0.219048
$682$		0			0
$683$	952.232	+	952.232i	1.39419	+	1.39419i	0.815664	+	0.578525i	$0.196372\pi$
$683$	952.232	+	952.232i	1.39419	+	1.39419i	0.578525	+	0.815664i	$0.303628\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−169.955	+	169.955i	−0.247387	+	0.247387i
$688$		0			0
$689$			255.192i			0.370380i
$690$		0			0
$691$	−568.665			−0.822960			−0.411480	−	0.911419i	$-0.634988\pi$
$691$	−568.665			−0.822960			−0.411480	+	0.911419i	$0.634988\pi$
$692$		0			0
$693$	−228.272	−	228.272i	−0.329397	−	0.329397i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	54.6469	−	54.6469i	0.0784031	−	0.0784031i
$698$		0			0
$699$			623.778i			0.892386i
$700$		0			0
$701$	−125.698			−0.179312			−0.0896562	−	0.995973i	$-0.528577\pi$
$701$	−125.698			−0.179312			−0.0896562	+	0.995973i	$0.528577\pi$
$702$		0			0
$703$	−307.283	−	307.283i	−0.437102	−	0.437102i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−1000.91	+	1000.91i	−1.41571	+	1.41571i
$708$		0			0
$709$			201.069i			0.283596i	0.989896	+	0.141798i	$0.0452883\pi$
$709$			201.069i			0.283596i	−0.989896	+	0.141798i	$0.954712\pi$
$710$		0			0
$711$	198.000			0.278481
$712$		0			0
$713$	−549.131	−	549.131i	−0.770169	−	0.770169i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−517.807	+	517.807i	−0.722186	+	0.722186i
$718$		0			0
$719$			785.576i			1.09259i	0.837591	+	0.546297i	$0.183963\pi$
$719$			785.576i			1.09259i	−0.837591	+	0.546297i	$0.816037\pi$
$720$		0			0
$721$	−546.384			−0.757814
$722$		0			0
$723$	13.1872	+	13.1872i	0.0182396	+	0.0182396i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−282.459	+	282.459i	−0.388526	+	0.388526i	−0.874162	−	0.485635i	$-0.838588\pi$
$727$	−282.459	+	282.459i	−0.388526	+	0.388526i	0.485635	+	0.874162i	$0.338588\pi$
$728$		0			0
$729$		−	27.0000i		−	0.0370370i
$730$		0			0
$731$	−39.9184			−0.0546079
$732$		0			0
$733$	−54.9036	−	54.9036i	−0.0749025	−	0.0749025i	0.668663	−	0.743566i	$-0.266867\pi$
$733$	−54.9036	−	54.9036i	−0.0749025	−	0.0749025i	−0.743566	+	0.668663i	$0.766867\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−918.929	+	918.929i	−1.24685	+	1.24685i
$738$		0			0
$739$		−	276.624i		−	0.374323i	−0.982329	−	0.187161i	$-0.940071\pi$
$739$		−	276.624i		−	0.374323i	0.982329	−	0.187161i	$-0.0599287\pi$
$740$		0			0
$741$	201.576			0.272032
$742$		0			0
$743$	−143.101	−	143.101i	−0.192599	−	0.192599i	0.604219	−	0.796818i	$-0.293485\pi$
$743$	−143.101	−	143.101i	−0.192599	−	0.192599i	−0.796818	+	0.604219i	$0.793485\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	252.272	−	252.272i	0.337714	−	0.337714i
$748$		0			0
$749$		−	385.535i		−	0.514733i
$750$		0			0
$751$	−871.453			−1.16039			−0.580195	−	0.814478i	$-0.697024\pi$
$751$	−871.453			−1.16039			−0.580195	+	0.814478i	$0.697024\pi$
$752$		0			0
$753$	217.782	+	217.782i	0.289219	+	0.289219i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−304.702	+	304.702i	−0.402512	+	0.402512i	−0.879117	−	0.476605i	$-0.841867\pi$
$757$	−304.702	+	304.702i	−0.402512	+	0.402512i	0.476605	+	0.879117i	$0.341867\pi$
$758$		0			0
$759$		−	629.939i		−	0.829959i
$760$		0			0
$761$	613.959			0.806779			0.403390	−	0.915028i	$-0.367832\pi$
$761$	613.959			0.806779			0.403390	+	0.915028i	$0.367832\pi$
$762$		0			0
$763$	−110.879	−	110.879i	−0.145319	−	0.145319i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	213.081	−	213.081i	0.277810	−	0.277810i
$768$		0			0
$769$		−	887.412i		−	1.15398i	−0.816750	−	0.576991i	$-0.804227\pi$
$769$		−	887.412i		−	1.15398i	0.816750	−	0.576991i	$-0.195773\pi$
$770$		0			0
$771$	348.545			0.452069
$772$		0			0
$773$	−605.757	−	605.757i	−0.783644	−	0.783644i	0.196799	−	0.980444i	$-0.436945\pi$
$773$	−605.757	−	605.757i	−0.783644	−	0.783644i	−0.980444	+	0.196799i	$0.936945\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	144.495	−	144.495i	0.185965	−	0.185965i
$778$		0			0
$779$		−	2042.22i		−	2.62159i
$780$		0			0
$781$	787.959			1.00891
$782$		0			0
$783$	13.2122	+	13.2122i	0.0168739	+	0.0168739i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−616.463	+	616.463i	−0.783308	+	0.783308i	−0.980387	−	0.197080i	$-0.936854\pi$
$787$	−616.463	+	616.463i	−0.783308	+	0.783308i	0.197080	+	0.980387i	$0.436854\pi$
$788$		0			0
$789$		−	88.1816i		−	0.111764i
$790$		0			0
$791$	−1732.24			−2.18994
$792$		0			0
$793$	265.485	+	265.485i	0.334785	+	0.334785i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	247.768	−	247.768i	0.310876	−	0.310876i	−0.534373	−	0.845249i	$-0.679452\pi$
$797$	247.768	−	247.768i	0.310876	−	0.310876i	0.845249	+	0.534373i	$0.179452\pi$
$798$		0			0
$799$		−	8.08164i		−	0.0101147i
$800$		0			0
$801$	28.7878			0.0359398
$802$		0			0
$803$	−928.463	−	928.463i	−1.15624	−	1.15624i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−47.5551	+	47.5551i	−0.0589282	+	0.0589282i
$808$		0			0
$809$			1556.10i			1.92348i	0.273958	+	0.961742i	$0.411667\pi$
$809$			1556.10i			1.92348i	−0.273958	+	0.961742i	$0.588333\pi$
$810$		0			0
$811$	−277.192			−0.341790			−0.170895	−	0.985289i	$-0.554666\pi$
$811$	−277.192			−0.341790			−0.170895	+	0.985289i	$0.554666\pi$
$812$		0			0
$813$	14.1770	+	14.1770i	0.0174379	+	0.0174379i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−745.898	+	745.898i	−0.912972	+	0.912972i
$818$		0			0
$819$			94.7878i			0.115736i
$820$		0			0
$821$	−1485.86			−1.80981			−0.904907	−	0.425610i	$-0.860060\pi$
$821$	−1485.86			−1.80981			−0.904907	+	0.425610i	$0.860060\pi$
$822$		0			0
$823$	−809.519	−	809.519i	−0.983620	−	0.983620i	0.0162485	−	0.999868i	$-0.494828\pi$
$823$	−809.519	−	809.519i	−0.983620	−	0.983620i	−0.999868	+	0.0162485i	$0.994828\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	1122.66	−	1122.66i	1.35750	−	1.35750i	0.480521	−	0.876983i	$-0.340447\pi$
$827$	1122.66	−	1122.66i	1.35750	−	1.35750i	0.876983	−	0.480521i	$-0.159553\pi$
$828$		0			0
$829$			849.151i			1.02431i	0.858894	+	0.512154i	$0.171152\pi$
$829$			849.151i			1.02431i	−0.858894	+	0.512154i	$0.828848\pi$
$830$		0			0
$831$	691.980			0.832707
$832$		0			0
$833$	30.7378	+	30.7378i	0.0369001	+	0.0369001i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−92.5607	+	92.5607i	−0.110586	+	0.110586i
$838$		0			0
$839$			1606.26i			1.91449i	0.289270	+	0.957247i	$0.406587\pi$
$839$			1606.26i			1.91449i	−0.289270	+	0.957247i	$0.593413\pi$
$840$		0			0
$841$	828.069			0.984625
$842$		0			0
$843$	529.560	+	529.560i	0.628185	+	0.628185i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−117.328	+	117.328i	−0.138522	+	0.138522i
$848$		0			0
$849$			548.363i			0.645893i
$850$		0			0
$851$	398.747			0.468563
$852$		0			0
$853$	−632.702	−	632.702i	−0.741737	−	0.741737i	0.231175	−	0.972912i	$-0.425743\pi$
$853$	−632.702	−	632.702i	−0.741737	−	0.741737i	−0.972912	+	0.231175i	$0.925743\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−258.879	+	258.879i	−0.302075	+	0.302075i	−0.841825	−	0.539750i	$-0.818519\pi$
$857$	−258.879	+	258.879i	−0.302075	+	0.302075i	0.539750	+	0.841825i	$0.318519\pi$
$858$		0			0
$859$			534.082i			0.621748i	0.950451	+	0.310874i	$0.100622\pi$
$859$			534.082i			0.621748i	−0.950451	+	0.310874i	$0.899378\pi$
$860$		0			0
$861$	960.322			1.11536
$862$		0			0
$863$	−711.878	−	711.878i	−0.824887	−	0.824887i	0.161917	−	0.986804i	$-0.448232\pi$
$863$	−711.878	−	711.878i	−0.824887	−	0.824887i	−0.986804	+	0.161917i	$0.948232\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−351.972	+	351.972i	−0.405965	+	0.405965i
$868$		0			0
$869$		−	778.665i		−	0.896048i
$870$		0			0
$871$	381.576			0.438089
$872$		0			0
$873$	−159.242	−	159.242i	−0.182408	−	0.182408i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−742.590	+	742.590i	−0.846739	+	0.846739i	−0.989725	−	0.142986i	$-0.954330\pi$
$877$	−742.590	+	742.590i	−0.846739	+	0.846739i	0.142986	+	0.989725i	$0.454330\pi$
$878$		0			0
$879$		−	481.980i		−	0.548327i
$880$		0			0
$881$	−1000.75			−1.13592			−0.567961	−	0.823056i	$-0.692267\pi$
$881$	−1000.75			−1.13592			−0.567961	+	0.823056i	$0.692267\pi$
$882$		0			0
$883$	546.788	+	546.788i	0.619239	+	0.619239i	0.945336	−	0.326097i	$-0.105734\pi$
$883$	546.788	+	546.788i	0.619239	+	0.619239i	−0.326097	+	0.945336i	$0.605734\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−1138.48	+	1138.48i	−1.28352	+	1.28352i	−0.344872	+	0.938650i	$0.612078\pi$
$887$	−1138.48	+	1138.48i	−1.28352	+	1.28352i	−0.938650	+	0.344872i	$0.887922\pi$
$888$		0			0
$889$		−	1976.60i		−	2.22340i
$890$		0			0
$891$	−106.182			−0.119171
$892$		0			0
$893$	−151.010	−	151.010i	−0.169104	−	0.169104i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−130.788	+	130.788i	−0.145806	+	0.145806i
$898$		0			0
$899$		−	90.5878i		−	0.100765i
$900$		0			0
$901$	93.6571			0.103948
$902$		0			0
$903$	−350.747	−	350.747i	−0.388424	−	0.388424i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	162.706	−	162.706i	0.179389	−	0.179389i	−0.611700	−	0.791090i	$-0.709514\pi$
$907$	162.706	−	162.706i	0.179389	−	0.179389i	0.791090	+	0.611700i	$0.209514\pi$
$908$		0			0
$909$			465.576i			0.512184i
$910$		0			0
$911$	386.988			0.424794			0.212397	−	0.977183i	$-0.431873\pi$
$911$	386.988			0.424794			0.212397	+	0.977183i	$0.431873\pi$
$912$		0			0
$913$	−992.100	−	992.100i	−1.08664	−	1.08664i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−1268.01	+	1268.01i	−1.38278	+	1.38278i
$918$		0			0
$919$			1101.80i			1.19891i	0.800409	+	0.599454i	$0.204615\pi$
$919$			1101.80i			1.19891i	−0.800409	+	0.599454i	$0.795385\pi$
$920$		0			0
$921$	−347.110			−0.376884
$922$		0			0
$923$	−163.596	−	163.596i	−0.177244	−	0.177244i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−127.076	+	127.076i	−0.137083	+	0.137083i
$928$		0			0
$929$		−	261.273i		−	0.281242i	−0.990064	−	0.140621i	$-0.955090\pi$
$929$		−	261.273i		−	0.281242i	0.990064	−	0.140621i	$-0.0449098\pi$
$930$		0			0
$931$	1148.71			1.23384
$932$		0			0
$933$	504.050	+	504.050i	0.540247	+	0.540247i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	57.7980	−	57.7980i	0.0616841	−	0.0616841i	−0.675592	−	0.737276i	$-0.736112\pi$
$937$	57.7980	−	57.7980i	0.0616841	−	0.0616841i	0.737276	+	0.675592i	$0.236112\pi$
$938$		0			0
$939$			896.363i			0.954593i
$940$		0			0
$941$	399.837			0.424906			0.212453	−	0.977171i	$-0.431855\pi$
$941$	399.837			0.424906			0.212453	+	0.977171i	$0.431855\pi$
$942$		0			0
$943$	1325.05	+	1325.05i	1.40514	+	1.40514i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−464.958	+	464.958i	−0.490980	+	0.490980i	−0.908615	−	0.417635i	$-0.862859\pi$
$947$	−464.958	+	464.958i	−0.490980	+	0.490980i	0.417635	+	0.908615i	$0.362859\pi$
$948$		0			0
$949$			385.535i			0.406254i
$950$		0			0
$951$	828.767			0.871469
$952$		0			0
$953$	−627.242	−	627.242i	−0.658176	−	0.658176i	0.296772	−	0.954948i	$-0.404090\pi$
$953$	−627.242	−	627.242i	−0.658176	−	0.658176i	−0.954948	+	0.296772i	$0.904090\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	51.9592	−	51.9592i	0.0542938	−	0.0542938i
$958$		0			0
$959$		−	1045.86i		−	1.09057i
$960$		0			0
$961$	−326.371			−0.339616
$962$		0			0
$963$	−89.6663	−	89.6663i	−0.0931114	−	0.0931114i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−1334.02	+	1334.02i	−1.37955	+	1.37955i	−0.534173	+	0.845375i	$0.679377\pi$
$967$	−1334.02	+	1334.02i	−1.37955	+	1.37955i	−0.845375	+	0.534173i	$0.820623\pi$
$968$		0			0
$969$		−	73.9796i		−	0.0763463i
$970$		0			0
$971$	1489.17			1.53365			0.766823	−	0.641859i	$-0.221837\pi$
$971$	1489.17			1.53365			0.766823	+	0.641859i	$0.221837\pi$
$972$		0			0
$973$	239.868	+	239.868i	0.246525	+	0.246525i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1134.66	−	1134.66i	1.16137	−	1.16137i	0.177191	−	0.984176i	$-0.443299\pi$
$977$	1134.66	−	1134.66i	1.16137	−	1.16137i	0.984176	−	0.177191i	$-0.0567011\pi$
$978$		0			0
$979$		−	113.212i		−	0.115641i
$980$		0			0
$981$	−51.5755			−0.0525744
$982$		0			0
$983$	−261.485	−	261.485i	−0.266007	−	0.266007i	0.561482	−	0.827489i	$-0.310231\pi$
$983$	−261.485	−	261.485i	−0.266007	−	0.266007i	−0.827489	+	0.561482i	$0.810231\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	71.0102	−	71.0102i	0.0719455	−	0.0719455i
$988$		0			0
$989$		−	967.918i		−	0.978684i
$990$		0			0
$991$	950.645			0.959278			0.479639	−	0.877466i	$-0.340768\pi$
$991$	950.645			0.959278			0.479639	+	0.877466i	$0.340768\pi$
$992$		0			0
$993$	119.480	+	119.480i	0.120322	+	0.120322i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	96.0566	−	96.0566i	0.0963457	−	0.0963457i	−0.657291	−	0.753637i	$-0.728298\pi$
$997$	96.0566	−	96.0566i	0.0963457	−	0.0963457i	0.753637	+	0.657291i	$0.228298\pi$
$998$		0			0
$999$		−	67.2122i		−	0.0672795i

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1200.3.bg.b.193.2		4
4.3	odd	2		600.3.u.f.193.1	yes	4
5.2	odd	4	inner	1200.3.bg.b.1057.2		4
5.3	odd	4		1200.3.bg.m.1057.1		4
5.4	even	2		1200.3.bg.m.193.1		4
12.11	even	2		1800.3.v.q.793.2		4
20.3	even	4		600.3.u.a.457.2	yes	4
20.7	even	4		600.3.u.f.457.1	yes	4
20.19	odd	2		600.3.u.a.193.2	✓	4
60.23	odd	4		1800.3.v.j.1657.1		4
60.47	odd	4		1800.3.v.q.1657.2		4
60.59	even	2		1800.3.v.j.793.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
600.3.u.a.193.2	✓	4	20.19	odd	2
600.3.u.a.457.2	yes	4	20.3	even	4
600.3.u.f.193.1	yes	4	4.3	odd	2
600.3.u.f.457.1	yes	4	20.7	even	4
1200.3.bg.b.193.2		4	1.1	even	1	trivial
1200.3.bg.b.1057.2		4	5.2	odd	4	inner
1200.3.bg.m.193.1		4	5.4	even	2
1200.3.bg.m.1057.1		4	5.3	odd	4
1800.3.v.j.793.1		4	60.59	even	2
1800.3.v.j.1657.1		4	60.23	odd	4
1800.3.v.q.793.2		4	12.11	even	2
1800.3.v.q.1657.2		4	60.47	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	1200.3.bg.b.193.2

Level	$1200$
Weight	$3$
Character	1200.193
Analytic conductor	$32.698$
Analytic rank	$0$
Dimension	$4$
Inner twists	$2$